Greatest common factor

One of the more useful techniques for simplifying expressions and numbers is finding the greatest common factor. The largest number (factor) that divides two or more numbers is called the greatest common factor or GCF.

Two methods are used to find the greatest common factor. The first method includes writing down all the factors of two or more numbers. After that, we need to write all the common factors for each number. The greatest common factor of the numbers is the largest number in that list.

First GCF method

Let’s try this method on the next example. We need to find the largest common factor for numbers and .
First, we need to list all the factors for each number. In order to find all the factors of the number , we need to see if the number can be divided by a number between and . If numbr can be divided by that number without leaving a remainder, then that number is one of the factors of the number .
-> Number can be divided by so one of the factors is .
-> Number can be divided by so one of the factors is .
-> Number can be divided by so one of the factors is .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by and .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by and .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by any number between and .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by any number between and .
-> Number can be divided by and this is the last factor.

We need to list the factors. They are and .

We need to repeat the process for number :
-> Number can be divided by so one of the factors is .
-> Number can be divided by so one of the factors is .
-> Number can be divided by so one of the factors is .
-> Number can be divided by 4 so one of the factors is .
-> Number can’t be divided by .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by , or .
-> Number can be divided by so one of the factors is .
-> Number can’t be divided by any number between and .
-> Number can be divided by 24 and this is the last factor.
Now we need to list these factors. They are: and .
Let’s compare them to the factors of : and .
Now, we need to list the common factors for these numbers and pick the highest one: .
We can see that the greatest common factor is of and is .

Second GCF method

The second method for finding the greatest common factor is to list all the prime factors for the two numbers. After that, we need to multiply the common prime numbers and we will get the greatest common factor.
Let’s try this method out with the same numbers from the previous example.

First we need to list the prime factors of .

To find the prime factors, we need to start dividing with the lowest possible number that can divide .

We can start with number . Number can divide number and the result is . That means that number is the first prime factor.
-> Number can divide and the result is . That means that the second prime factor is .
-> Number can’t divide . The next number that we should try is number . Number can divide number . The result is number , so is the third prime factor.
-> Number can divide . The result is . The fourth prime factor is number .
-> Since the last result is , that means that the calculation of the prime factors is done.
-> The prime factors of number are:

Using the same method, we can find the prime factors for number .

The prime factors of are:
The prime factors of number are:
Now we need to pick the common prime factors. We can see that the common prime factors are and . When we multiply we get which is the greatest common factor.